--- a/CpsG.thy_1_1 Fri Jan 29 17:06:02 2016 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1751 +0,0 @@
-theory CpsG
-imports PIPDefs
-begin
-
-lemma Max_f_mono:
- assumes seq: "A \<subseteq> B"
- and np: "A \<noteq> {}"
- and fnt: "finite B"
- shows "Max (f ` A) \<le> Max (f ` B)"
-proof(rule Max_mono)
- from seq show "f ` A \<subseteq> f ` B" by auto
-next
- from np show "f ` A \<noteq> {}" by auto
-next
- from fnt and seq show "finite (f ` B)" by auto
-qed
-
-
-locale valid_trace =
- fixes s
- assumes vt : "vt s"
-
-locale valid_trace_e = valid_trace +
- fixes e
- assumes vt_e: "vt (e#s)"
-begin
-
-lemma pip_e: "PIP s e"
- using vt_e by (cases, simp)
-
-end
-
-locale valid_trace_create = valid_trace_e +
- fixes th prio
- assumes is_create: "e = Create th prio"
-
-locale valid_trace_exit = valid_trace_e +
- fixes th
- assumes is_exit: "e = Exit th"
-
-locale valid_trace_p = valid_trace_e +
- fixes th cs
- assumes is_p: "e = P th cs"
-
-locale valid_trace_v = valid_trace_e +
- fixes th cs
- assumes is_v: "e = V th cs"
-begin
- definition "rest = tl (wq s cs)"
- definition "wq' = (SOME q. distinct q \<and> set q = set rest)"
-end
-
-locale valid_trace_v_n = valid_trace_v +
- assumes rest_nnl: "rest \<noteq> []"
-
-locale valid_trace_v_e = valid_trace_v +
- assumes rest_nil: "rest = []"
-
-locale valid_trace_set= valid_trace_e +
- fixes th prio
- assumes is_set: "e = Set th prio"
-
-context valid_trace
-begin
-
-lemma ind [consumes 0, case_names Nil Cons, induct type]:
- assumes "PP []"
- and "(\<And>s e. valid_trace_e s e \<Longrightarrow>
- PP s \<Longrightarrow> PIP s e \<Longrightarrow> PP (e # s))"
- shows "PP s"
-proof(induct rule:vt.induct[OF vt, case_names Init Step])
- case Init
- from assms(1) show ?case .
-next
- case (Step s e)
- show ?case
- proof(rule assms(2))
- show "valid_trace_e s e" using Step by (unfold_locales, auto)
- next
- show "PP s" using Step by simp
- next
- show "PIP s e" using Step by simp
- qed
-qed
-
-lemma vt_moment: "\<And> t. vt (moment t s)"
-proof(induct rule:ind)
- case Nil
- thus ?case by (simp add:vt_nil)
-next
- case (Cons s e t)
- show ?case
- proof(cases "t \<ge> length (e#s)")
- case True
- from True have "moment t (e#s) = e#s" by simp
- thus ?thesis using Cons
- by (simp add:valid_trace_def valid_trace_e_def, auto)
- next
- case False
- from Cons have "vt (moment t s)" by simp
- moreover have "moment t (e#s) = moment t s"
- proof -
- from False have "t \<le> length s" by simp
- from moment_app [OF this, of "[e]"]
- show ?thesis by simp
- qed
- ultimately show ?thesis by simp
- qed
-qed
-
-lemma finite_threads:
- shows "finite (threads s)"
-using vt by (induct) (auto elim: step.cases)
-
-end
-
-lemma cp_eq_cpreced: "cp s th = cpreced (wq s) s th"
-unfolding cp_def wq_def
-apply(induct s rule: schs.induct)
-apply(simp add: Let_def cpreced_initial)
-apply(simp add: Let_def)
-apply(simp add: Let_def)
-apply(simp add: Let_def)
-apply(subst (2) schs.simps)
-apply(simp add: Let_def)
-apply(subst (2) schs.simps)
-apply(simp add: Let_def)
-done
-
-lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
- by (unfold s_RAG_def, auto)
-
-locale valid_moment = valid_trace +
- fixes i :: nat
-
-sublocale valid_moment < vat_moment: valid_trace "(moment i s)"
- by (unfold_locales, insert vt_moment, auto)
-
-lemma waiting_eq: "waiting s th cs = waiting (wq s) th cs"
- by (unfold s_waiting_def cs_waiting_def wq_def, auto)
-
-lemma holding_eq: "holding (s::state) th cs = holding (wq s) th cs"
- by (unfold s_holding_def wq_def cs_holding_def, simp)
-
-lemma runing_ready:
- shows "runing s \<subseteq> readys s"
- unfolding runing_def readys_def
- by auto
-
-lemma readys_threads:
- shows "readys s \<subseteq> threads s"
- unfolding readys_def
- by auto
-
-lemma wq_v_neq [simp]:
- "cs \<noteq> cs' \<Longrightarrow> wq (V thread cs#s) cs' = wq s cs'"
- by (auto simp:wq_def Let_def cp_def split:list.splits)
-
-lemma runing_head:
- assumes "th \<in> runing s"
- and "th \<in> set (wq_fun (schs s) cs)"
- shows "th = hd (wq_fun (schs s) cs)"
- using assms
- by (simp add:runing_def readys_def s_waiting_def wq_def)
-
-context valid_trace
-begin
-
-lemma runing_wqE:
- assumes "th \<in> runing s"
- and "th \<in> set (wq s cs)"
- obtains rest where "wq s cs = th#rest"
-proof -
- from assms(2) obtain th' rest where eq_wq: "wq s cs = th'#rest"
- by (meson list.set_cases)
- have "th' = th"
- proof(rule ccontr)
- assume "th' \<noteq> th"
- hence "th \<noteq> hd (wq s cs)" using eq_wq by auto
- with assms(2)
- have "waiting s th cs"
- by (unfold s_waiting_def, fold wq_def, auto)
- with assms show False
- by (unfold runing_def readys_def, auto)
- qed
- with eq_wq that show ?thesis by metis
-qed
-
-end
-
-context valid_trace_create
-begin
-
-lemma wq_neq_simp [simp]:
- shows "wq (e#s) cs' = wq s cs'"
- using assms unfolding is_create wq_def
- by (auto simp:Let_def)
-
-lemma wq_distinct_kept:
- assumes "distinct (wq s cs')"
- shows "distinct (wq (e#s) cs')"
- using assms by simp
-end
-
-context valid_trace_exit
-begin
-
-lemma wq_neq_simp [simp]:
- shows "wq (e#s) cs' = wq s cs'"
- using assms unfolding is_exit wq_def
- by (auto simp:Let_def)
-
-lemma wq_distinct_kept:
- assumes "distinct (wq s cs')"
- shows "distinct (wq (e#s) cs')"
- using assms by simp
-end
-
-context valid_trace_p
-begin
-
-lemma wq_neq_simp [simp]:
- assumes "cs' \<noteq> cs"
- shows "wq (e#s) cs' = wq s cs'"
- using assms unfolding is_p wq_def
- by (auto simp:Let_def)
-
-lemma runing_th_s:
- shows "th \<in> runing s"
-proof -
- from pip_e[unfolded is_p]
- show ?thesis by (cases, simp)
-qed
-
-lemma th_not_waiting:
- "\<not> waiting s th c"
-proof -
- have "th \<in> readys s"
- using runing_ready runing_th_s by blast
- thus ?thesis
- by (unfold readys_def, auto)
-qed
-
-lemma waiting_neq_th:
- assumes "waiting s t c"
- shows "t \<noteq> th"
- using assms using th_not_waiting by blast
-
-lemma th_not_in_wq:
- shows "th \<notin> set (wq s cs)"
-proof
- assume otherwise: "th \<in> set (wq s cs)"
- from runing_wqE[OF runing_th_s this]
- obtain rest where eq_wq: "wq s cs = th#rest" by blast
- with otherwise
- have "holding s th cs"
- by (unfold s_holding_def, fold wq_def, simp)
- hence cs_th_RAG: "(Cs cs, Th th) \<in> RAG s"
- by (unfold s_RAG_def, fold holding_eq, auto)
- from pip_e[unfolded is_p]
- show False
- proof(cases)
- case (thread_P)
- with cs_th_RAG show ?thesis by auto
- qed
-qed
-
-lemma wq_es_cs:
- "wq (e#s) cs = wq s cs @ [th]"
- by (unfold is_p wq_def, auto simp:Let_def)
-
-lemma wq_distinct_kept:
- assumes "distinct (wq s cs')"
- shows "distinct (wq (e#s) cs')"
-proof(cases "cs' = cs")
- case True
- show ?thesis using True assms th_not_in_wq
- by (unfold True wq_es_cs, auto)
-qed (insert assms, simp)
-
-end
-
-context valid_trace_v
-begin
-
-lemma wq_neq_simp [simp]:
- assumes "cs' \<noteq> cs"
- shows "wq (e#s) cs' = wq s cs'"
- using assms unfolding is_v wq_def
- by (auto simp:Let_def)
-
-lemma runing_th_s:
- shows "th \<in> runing s"
-proof -
- from pip_e[unfolded is_v]
- show ?thesis by (cases, simp)
-qed
-
-lemma th_not_waiting:
- "\<not> waiting s th c"
-proof -
- have "th \<in> readys s"
- using runing_ready runing_th_s by blast
- thus ?thesis
- by (unfold readys_def, auto)
-qed
-
-lemma waiting_neq_th:
- assumes "waiting s t c"
- shows "t \<noteq> th"
- using assms using th_not_waiting by blast
-
-lemma wq_s_cs:
- "wq s cs = th#rest"
-proof -
- from pip_e[unfolded is_v]
- show ?thesis
- proof(cases)
- case (thread_V)
- from this(2) show ?thesis
- by (unfold rest_def s_holding_def, fold wq_def,
- metis empty_iff list.collapse list.set(1))
- qed
-qed
-
-lemma wq_es_cs:
- "wq (e#s) cs = wq'"
- using wq_s_cs[unfolded wq_def]
- by (auto simp:Let_def wq_def rest_def wq'_def is_v, simp)
-
-lemma wq_distinct_kept:
- assumes "distinct (wq s cs')"
- shows "distinct (wq (e#s) cs')"
-proof(cases "cs' = cs")
- case True
- show ?thesis
- proof(unfold True wq_es_cs wq'_def, rule someI2)
- show "distinct rest \<and> set rest = set rest"
- using assms[unfolded True wq_s_cs] by auto
- qed simp
-qed (insert assms, simp)
-
-end
-
-context valid_trace_set
-begin
-
-lemma wq_neq_simp [simp]:
- shows "wq (e#s) cs' = wq s cs'"
- using assms unfolding is_set wq_def
- by (auto simp:Let_def)
-
-lemma wq_distinct_kept:
- assumes "distinct (wq s cs')"
- shows "distinct (wq (e#s) cs')"
- using assms by simp
-end
-
-context valid_trace
-begin
-
-lemma actor_inv:
- assumes "PIP s e"
- and "\<not> isCreate e"
- shows "actor e \<in> runing s"
- using assms
- by (induct, auto)
-
-lemma isP_E:
- assumes "isP e"
- obtains cs where "e = P (actor e) cs"
- using assms by (cases e, auto)
-
-lemma isV_E:
- assumes "isV e"
- obtains cs where "e = V (actor e) cs"
- using assms by (cases e, auto)
-
-lemma wq_distinct: "distinct (wq s cs)"
-proof(induct rule:ind)
- case (Cons s e)
- interpret vt_e: valid_trace_e s e using Cons by simp
- show ?case
- proof(cases e)
- case (Create th prio)
- interpret vt_create: valid_trace_create s e th prio
- using Create by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt_create.wq_distinct_kept)
- next
- case (Exit th)
- interpret vt_exit: valid_trace_exit s e th
- using Exit by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt_exit.wq_distinct_kept)
- next
- case (P th cs)
- interpret vt_p: valid_trace_p s e th cs using P by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt_p.wq_distinct_kept)
- next
- case (V th cs)
- interpret vt_v: valid_trace_v s e th cs using V by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt_v.wq_distinct_kept)
- next
- case (Set th prio)
- interpret vt_set: valid_trace_set s e th prio
- using Set by (unfold_locales, simp)
- show ?thesis using Cons by (simp add: vt_set.wq_distinct_kept)
- qed
-qed (unfold wq_def Let_def, simp)
-
-end
-
-context valid_trace_e
-begin
-
-text {*
- The following lemma shows that only the @{text "P"}
- operation can add new thread into waiting queues.
- Such kind of lemmas are very obvious, but need to be checked formally.
- This is a kind of confirmation that our modelling is correct.
-*}
-
-lemma wq_in_inv:
- assumes s_ni: "thread \<notin> set (wq s cs)"
- and s_i: "thread \<in> set (wq (e#s) cs)"
- shows "e = P thread cs"
-proof(cases e)
- -- {* This is the only non-trivial case: *}
- case (V th cs1)
- have False
- proof(cases "cs1 = cs")
- case True
- show ?thesis
- proof(cases "(wq s cs1)")
- case (Cons w_hd w_tl)
- have "set (wq (e#s) cs) \<subseteq> set (wq s cs)"
- proof -
- have "(wq (e#s) cs) = (SOME q. distinct q \<and> set q = set w_tl)"
- using Cons V by (auto simp:wq_def Let_def True split:if_splits)
- moreover have "set ... \<subseteq> set (wq s cs)"
- proof(rule someI2)
- show "distinct w_tl \<and> set w_tl = set w_tl"
- by (metis distinct.simps(2) local.Cons wq_distinct)
- qed (insert Cons True, auto)
- ultimately show ?thesis by simp
- qed
- with assms show ?thesis by auto
- qed (insert assms V True, auto simp:wq_def Let_def split:if_splits)
- qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
- thus ?thesis by auto
-qed (insert assms, auto simp:wq_def Let_def split:if_splits)
-
-lemma wq_out_inv:
- assumes s_in: "thread \<in> set (wq s cs)"
- and s_hd: "thread = hd (wq s cs)"
- and s_i: "thread \<noteq> hd (wq (e#s) cs)"
- shows "e = V thread cs"
-proof(cases e)
--- {* There are only two non-trivial cases: *}
- case (V th cs1)
- show ?thesis
- proof(cases "cs1 = cs")
- case True
- have "PIP s (V th cs)" using pip_e[unfolded V[unfolded True]] .
- thus ?thesis
- proof(cases)
- case (thread_V)
- moreover have "th = thread" using thread_V(2) s_hd
- by (unfold s_holding_def wq_def, simp)
- ultimately show ?thesis using V True by simp
- qed
- qed (insert assms V, auto simp:wq_def Let_def split:if_splits)
-next
- case (P th cs1)
- show ?thesis
- proof(cases "cs1 = cs")
- case True
- with P have "wq (e#s) cs = wq_fun (schs s) cs @ [th]"
- by (auto simp:wq_def Let_def split:if_splits)
- with s_i s_hd s_in have False
- by (metis empty_iff hd_append2 list.set(1) wq_def)
- thus ?thesis by simp
- qed (insert assms P, auto simp:wq_def Let_def split:if_splits)
-qed (insert assms, auto simp:wq_def Let_def split:if_splits)
-
-end
-
-
-context valid_trace
-begin
-
-
-text {* (* ddd *)
- The nature of the work is like this: since it starts from a very simple and basic
- model, even intuitively very `basic` and `obvious` properties need to derived from scratch.
- For instance, the fact
- that one thread can not be blocked by two critical resources at the same time
- is obvious, because only running threads can make new requests, if one is waiting for
- a critical resource and get blocked, it can not make another resource request and get
- blocked the second time (because it is not running).
-
- To derive this fact, one needs to prove by contraction and
- reason about time (or @{text "moement"}). The reasoning is based on a generic theorem
- named @{text "p_split"}, which is about status changing along the time axis. It says if
- a condition @{text "Q"} is @{text "True"} at a state @{text "s"},
- but it was @{text "False"} at the very beginning, then there must exits a moment @{text "t"}
- in the history of @{text "s"} (notice that @{text "s"} itself is essentially the history
- of events leading to it), such that @{text "Q"} switched
- from being @{text "False"} to @{text "True"} and kept being @{text "True"}
- till the last moment of @{text "s"}.
-
- Suppose a thread @{text "th"} is blocked
- on @{text "cs1"} and @{text "cs2"} in some state @{text "s"},
- since no thread is blocked at the very beginning, by applying
- @{text "p_split"} to these two blocking facts, there exist
- two moments @{text "t1"} and @{text "t2"} in @{text "s"}, such that
- @{text "th"} got blocked on @{text "cs1"} and @{text "cs2"}
- and kept on blocked on them respectively ever since.
-
- Without lost of generality, we assume @{text "t1"} is earlier than @{text "t2"}.
- However, since @{text "th"} was blocked ever since memonent @{text "t1"}, so it was still
- in blocked state at moment @{text "t2"} and could not
- make any request and get blocked the second time: Contradiction.
-*}
-
-lemma waiting_unique_pre: (* ddd *)
- assumes h11: "thread \<in> set (wq s cs1)"
- and h12: "thread \<noteq> hd (wq s cs1)"
- assumes h21: "thread \<in> set (wq s cs2)"
- and h22: "thread \<noteq> hd (wq s cs2)"
- and neq12: "cs1 \<noteq> cs2"
- shows "False"
-proof -
- let "?Q" = "\<lambda> cs s. thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs)"
- from h11 and h12 have q1: "?Q cs1 s" by simp
- from h21 and h22 have q2: "?Q cs2 s" by simp
- have nq1: "\<not> ?Q cs1 []" by (simp add:wq_def)
- have nq2: "\<not> ?Q cs2 []" by (simp add:wq_def)
- from p_split [of "?Q cs1", OF q1 nq1]
- obtain t1 where lt1: "t1 < length s"
- and np1: "\<not> ?Q cs1 (moment t1 s)"
- and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))" by auto
- from p_split [of "?Q cs2", OF q2 nq2]
- obtain t2 where lt2: "t2 < length s"
- and np2: "\<not> ?Q cs2 (moment t2 s)"
- and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))" by auto
- { fix s cs
- assume q: "?Q cs s"
- have "thread \<notin> runing s"
- proof
- assume "thread \<in> runing s"
- hence " \<forall>cs. \<not> (thread \<in> set (wq_fun (schs s) cs) \<and>
- thread \<noteq> hd (wq_fun (schs s) cs))"
- by (unfold runing_def s_waiting_def readys_def, auto)
- from this[rule_format, of cs] q
- show False by (simp add: wq_def)
- qed
- } note q_not_runing = this
- { fix t1 t2 cs1 cs2
- assume lt1: "t1 < length s"
- and np1: "\<not> ?Q cs1 (moment t1 s)"
- and nn1: "(\<forall>i'>t1. ?Q cs1 (moment i' s))"
- and lt2: "t2 < length s"
- and np2: "\<not> ?Q cs2 (moment t2 s)"
- and nn2: "(\<forall>i'>t2. ?Q cs2 (moment i' s))"
- and lt12: "t1 < t2"
- let ?t3 = "Suc t2"
- from lt2 have le_t3: "?t3 \<le> length s" by auto
- from moment_plus [OF this]
- obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
- have "t2 < ?t3" by simp
- from nn2 [rule_format, OF this] and eq_m
- have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
- h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
- have "vt (e#moment t2 s)"
- proof -
- from vt_moment
- have "vt (moment ?t3 s)" .
- with eq_m show ?thesis by simp
- qed
- then interpret vt_e: valid_trace_e "moment t2 s" "e"
- by (unfold_locales, auto, cases, simp)
- have ?thesis
- proof -
- have "thread \<in> runing (moment t2 s)"
- proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
- case True
- have "e = V thread cs2"
- proof -
- have eq_th: "thread = hd (wq (moment t2 s) cs2)"
- using True and np2 by auto
- from vt_e.wq_out_inv[OF True this h2]
- show ?thesis .
- qed
- thus ?thesis using vt_e.actor_inv[OF vt_e.pip_e] by auto
- next
- case False
- have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] .
- with vt_e.actor_inv[OF vt_e.pip_e]
- show ?thesis by auto
- qed
- moreover have "thread \<notin> runing (moment t2 s)"
- by (rule q_not_runing[OF nn1[rule_format, OF lt12]])
- ultimately show ?thesis by simp
- qed
- } note lt_case = this
- show ?thesis
- proof -
- { assume "t1 < t2"
- from lt_case[OF lt1 np1 nn1 lt2 np2 nn2 this]
- have ?thesis .
- } moreover {
- assume "t2 < t1"
- from lt_case[OF lt2 np2 nn2 lt1 np1 nn1 this]
- have ?thesis .
- } moreover {
- assume eq_12: "t1 = t2"
- let ?t3 = "Suc t2"
- from lt2 have le_t3: "?t3 \<le> length s" by auto
- from moment_plus [OF this]
- obtain e where eq_m: "moment ?t3 s = e#moment t2 s" by auto
- have lt_2: "t2 < ?t3" by simp
- from nn2 [rule_format, OF this] and eq_m
- have h1: "thread \<in> set (wq (e#moment t2 s) cs2)" and
- h2: "thread \<noteq> hd (wq (e#moment t2 s) cs2)" by auto
- from nn1[rule_format, OF lt_2[folded eq_12]] eq_m[folded eq_12]
- have g1: "thread \<in> set (wq (e#moment t1 s) cs1)" and
- g2: "thread \<noteq> hd (wq (e#moment t1 s) cs1)" by auto
- have "vt (e#moment t2 s)"
- proof -
- from vt_moment
- have "vt (moment ?t3 s)" .
- with eq_m show ?thesis by simp
- qed
- then interpret vt_e: valid_trace_e "moment t2 s" "e"
- by (unfold_locales, auto, cases, simp)
- have "e = V thread cs2 \<or> e = P thread cs2"
- proof(cases "thread \<in> set (wq (moment t2 s) cs2)")
- case True
- have "e = V thread cs2"
- proof -
- have eq_th: "thread = hd (wq (moment t2 s) cs2)"
- using True and np2 by auto
- from vt_e.wq_out_inv[OF True this h2]
- show ?thesis .
- qed
- thus ?thesis by auto
- next
- case False
- have "e = P thread cs2" using vt_e.wq_in_inv[OF False h1] .
- thus ?thesis by auto
- qed
- moreover have "e = V thread cs1 \<or> e = P thread cs1"
- proof(cases "thread \<in> set (wq (moment t1 s) cs1)")
- case True
- have eq_th: "thread = hd (wq (moment t1 s) cs1)"
- using True and np1 by auto
- from vt_e.wq_out_inv[folded eq_12, OF True this g2]
- have "e = V thread cs1" .
- thus ?thesis by auto
- next
- case False
- have "e = P thread cs1" using vt_e.wq_in_inv[folded eq_12, OF False g1] .
- thus ?thesis by auto
- qed
- ultimately have ?thesis using neq12 by auto
- } ultimately show ?thesis using nat_neq_iff by blast
- qed
-qed
-
-text {*
- This lemma is a simple corrolary of @{text "waiting_unique_pre"}.
-*}
-
-lemma waiting_unique:
- assumes "waiting s th cs1"
- and "waiting s th cs2"
- shows "cs1 = cs2"
- using waiting_unique_pre assms
- unfolding wq_def s_waiting_def
- by auto
-
-end
-
-(* not used *)
-text {*
- Every thread can only be blocked on one critical resource,
- symmetrically, every critical resource can only be held by one thread.
- This fact is much more easier according to our definition.
-*}
-lemma held_unique:
- assumes "holding (s::event list) th1 cs"
- and "holding s th2 cs"
- shows "th1 = th2"
- by (insert assms, unfold s_holding_def, auto)
-
-lemma last_set_lt: "th \<in> threads s \<Longrightarrow> last_set th s < length s"
- apply (induct s, auto)
- by (case_tac a, auto split:if_splits)
-
-lemma last_set_unique:
- "\<lbrakk>last_set th1 s = last_set th2 s; th1 \<in> threads s; th2 \<in> threads s\<rbrakk>
- \<Longrightarrow> th1 = th2"
- apply (induct s, auto)
- by (case_tac a, auto split:if_splits dest:last_set_lt)
-
-lemma preced_unique :
- assumes pcd_eq: "preced th1 s = preced th2 s"
- and th_in1: "th1 \<in> threads s"
- and th_in2: " th2 \<in> threads s"
- shows "th1 = th2"
-proof -
- from pcd_eq have "last_set th1 s = last_set th2 s" by (simp add:preced_def)
- from last_set_unique [OF this th_in1 th_in2]
- show ?thesis .
-qed
-
-lemma preced_linorder:
- assumes neq_12: "th1 \<noteq> th2"
- and th_in1: "th1 \<in> threads s"
- and th_in2: " th2 \<in> threads s"
- shows "preced th1 s < preced th2 s \<or> preced th1 s > preced th2 s"
-proof -
- from preced_unique [OF _ th_in1 th_in2] and neq_12
- have "preced th1 s \<noteq> preced th2 s" by auto
- thus ?thesis by auto
-qed
-
-text {*
- The following three lemmas show that @{text "RAG"} does not change
- by the happening of @{text "Set"}, @{text "Create"} and @{text "Exit"}
- events, respectively.
-*}
-
-lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s"
-apply (unfold s_RAG_def s_waiting_def wq_def)
-by (simp add:Let_def)
-
-
-context valid_trace_v
-begin
-
-lemma distinct_rest: "distinct rest"
- by (simp add: distinct_tl rest_def wq_distinct)
-
-lemma holding_cs_eq_th:
- assumes "holding s t cs"
- shows "t = th"
-proof -
- from pip_e[unfolded is_v]
- show ?thesis
- proof(cases)
- case (thread_V)
- from held_unique[OF this(2) assms]
- show ?thesis by simp
- qed
-qed
-
-lemma distinct_wq': "distinct wq'"
- by (metis (mono_tags, lifting) distinct_rest some_eq_ex wq'_def)
-
-lemma th'_in_inv:
- assumes "th' \<in> set wq'"
- shows "th' \<in> set rest"
- using assms
- by (metis (mono_tags, lifting) distinct.simps(2)
- rest_def some_eq_ex wq'_def wq_distinct wq_s_cs)
-
-lemma neq_t_th:
- assumes "waiting (e#s) t c"
- shows "t \<noteq> th"
-proof
- assume otherwise: "t = th"
- show False
- proof(cases "c = cs")
- case True
- have "t \<in> set wq'"
- using assms[unfolded True s_waiting_def, folded wq_def, unfolded wq_es_cs]
- by simp
- from th'_in_inv[OF this] have "t \<in> set rest" .
- with wq_s_cs[folded otherwise] wq_distinct[of cs]
- show ?thesis by simp
- next
- case False
- have "wq (e#s) c = wq s c" using False
- by (unfold is_v, simp)
- hence "waiting s t c" using assms
- by (simp add: cs_waiting_def waiting_eq)
- hence "t \<notin> readys s" by (unfold readys_def, auto)
- hence "t \<notin> runing s" using runing_ready by auto
- with runing_th_s[folded otherwise] show ?thesis by auto
- qed
-qed
-
-lemma waiting_esI1:
- assumes "waiting s t c"
- and "c \<noteq> cs"
- shows "waiting (e#s) t c"
-proof -
- have "wq (e#s) c = wq s c"
- using assms(2) is_v by auto
- with assms(1) show ?thesis
- using cs_waiting_def waiting_eq by auto
-qed
-
-lemma holding_esI2:
- assumes "c \<noteq> cs"
- and "holding s t c"
- shows "holding (e#s) t c"
-proof -
- from assms(1) have "wq (e#s) c = wq s c" using is_v by auto
- from assms(2)[unfolded s_holding_def, folded wq_def,
- folded this, unfolded wq_def, folded s_holding_def]
- show ?thesis .
-qed
-
-lemma holding_esI1:
- assumes "holding s t c"
- and "t \<noteq> th"
- shows "holding (e#s) t c"
-proof -
- have "c \<noteq> cs" using assms using holding_cs_eq_th by blast
- from holding_esI2[OF this assms(1)]
- show ?thesis .
-qed
-
-end
-
-context valid_trace_v_n
-begin
-
-lemma neq_wq': "wq' \<noteq> []"
-proof (unfold wq'_def, rule someI2)
- show "distinct rest \<and> set rest = set rest"
- by (simp add: distinct_rest)
-next
- fix x
- assume " distinct x \<and> set x = set rest"
- thus "x \<noteq> []" using rest_nnl by auto
-qed
-
-definition "taker = hd wq'"
-
-definition "rest' = tl wq'"
-
-lemma eq_wq': "wq' = taker # rest'"
- by (simp add: neq_wq' rest'_def taker_def)
-
-lemma next_th_taker:
- shows "next_th s th cs taker"
- using rest_nnl taker_def wq'_def wq_s_cs
- by (auto simp:next_th_def)
-
-lemma taker_unique:
- assumes "next_th s th cs taker'"
- shows "taker' = taker"
-proof -
- from assms
- obtain rest' where
- h: "wq s cs = th # rest'"
- "taker' = hd (SOME q. distinct q \<and> set q = set rest')"
- by (unfold next_th_def, auto)
- with wq_s_cs have "rest' = rest" by auto
- thus ?thesis using h(2) taker_def wq'_def by auto
-qed
-
-lemma waiting_set_eq:
- "{(Th th', Cs cs) |th'. next_th s th cs th'} = {(Th taker, Cs cs)}"
- by (smt all_not_in_conv bot.extremum insertI1 insert_subset
- mem_Collect_eq next_th_taker subsetI subset_antisym taker_def taker_unique)
-
-lemma holding_set_eq:
- "{(Cs cs, Th th') |th'. next_th s th cs th'} = {(Cs cs, Th taker)}"
- using next_th_taker taker_def waiting_set_eq
- by fastforce
-
-lemma holding_taker:
- shows "holding (e#s) taker cs"
- by (unfold s_holding_def, fold wq_def, unfold wq_es_cs,
- auto simp:neq_wq' taker_def)
-
-lemma waiting_esI2:
- assumes "waiting s t cs"
- and "t \<noteq> taker"
- shows "waiting (e#s) t cs"
-proof -
- have "t \<in> set wq'"
- proof(unfold wq'_def, rule someI2)
- show "distinct rest \<and> set rest = set rest"
- by (simp add: distinct_rest)
- next
- fix x
- assume "distinct x \<and> set x = set rest"
- moreover have "t \<in> set rest"
- using assms(1) cs_waiting_def waiting_eq wq_s_cs by auto
- ultimately show "t \<in> set x" by simp
- qed
- moreover have "t \<noteq> hd wq'"
- using assms(2) taker_def by auto
- ultimately show ?thesis
- by (unfold s_waiting_def, fold wq_def, unfold wq_es_cs, simp)
-qed
-
-lemma waiting_esE:
- assumes "waiting (e#s) t c"
- obtains "c \<noteq> cs" "waiting s t c"
- | "c = cs" "t \<noteq> taker" "waiting s t cs" "t \<in> set rest'"
-proof(cases "c = cs")
- case False
- hence "wq (e#s) c = wq s c" using is_v by auto
- with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto
- from that(1)[OF False this] show ?thesis .
-next
- case True
- from assms[unfolded s_waiting_def True, folded wq_def, unfolded wq_es_cs]
- have "t \<noteq> hd wq'" "t \<in> set wq'" by auto
- hence "t \<noteq> taker" by (simp add: taker_def)
- moreover hence "t \<noteq> th" using assms neq_t_th by blast
- moreover have "t \<in> set rest" by (simp add: `t \<in> set wq'` th'_in_inv)
- ultimately have "waiting s t cs"
- by (metis cs_waiting_def list.distinct(2) list.sel(1)
- list.set_sel(2) rest_def waiting_eq wq_s_cs)
- show ?thesis using that(2)
- using True `t \<in> set wq'` `t \<noteq> taker` `waiting s t cs` eq_wq' by auto
-qed
-
-lemma holding_esI1:
- assumes "c = cs"
- and "t = taker"
- shows "holding (e#s) t c"
- by (unfold assms, simp add: holding_taker)
-
-lemma holding_esE:
- assumes "holding (e#s) t c"
- obtains "c = cs" "t = taker"
- | "c \<noteq> cs" "holding s t c"
-proof(cases "c = cs")
- case True
- from assms[unfolded True, unfolded s_holding_def,
- folded wq_def, unfolded wq_es_cs]
- have "t = taker" by (simp add: taker_def)
- from that(1)[OF True this] show ?thesis .
-next
- case False
- hence "wq (e#s) c = wq s c" using is_v by auto
- from assms[unfolded s_holding_def, folded wq_def,
- unfolded this, unfolded wq_def, folded s_holding_def]
- have "holding s t c" .
- from that(2)[OF False this] show ?thesis .
-qed
-
-end
-
-
-context valid_trace_v_e
-begin
-
-lemma nil_wq': "wq' = []"
-proof (unfold wq'_def, rule someI2)
- show "distinct rest \<and> set rest = set rest"
- by (simp add: distinct_rest)
-next
- fix x
- assume " distinct x \<and> set x = set rest"
- thus "x = []" using rest_nil by auto
-qed
-
-lemma no_taker:
- assumes "next_th s th cs taker"
- shows "False"
-proof -
- from assms[unfolded next_th_def]
- obtain rest' where "wq s cs = th # rest'" "rest' \<noteq> []"
- by auto
- thus ?thesis using rest_def rest_nil by auto
-qed
-
-lemma waiting_set_eq:
- "{(Th th', Cs cs) |th'. next_th s th cs th'} = {}"
- using no_taker by auto
-
-lemma holding_set_eq:
- "{(Cs cs, Th th') |th'. next_th s th cs th'} = {}"
- using no_taker by auto
-
-lemma no_holding:
- assumes "holding (e#s) taker cs"
- shows False
-proof -
- from wq_es_cs[unfolded nil_wq']
- have " wq (e # s) cs = []" .
- from assms[unfolded s_holding_def, folded wq_def, unfolded this]
- show ?thesis by auto
-qed
-
-lemma no_waiting:
- assumes "waiting (e#s) t cs"
- shows False
-proof -
- from wq_es_cs[unfolded nil_wq']
- have " wq (e # s) cs = []" .
- from assms[unfolded s_waiting_def, folded wq_def, unfolded this]
- show ?thesis by auto
-qed
-
-lemma waiting_esI2:
- assumes "waiting s t c"
- shows "waiting (e#s) t c"
-proof -
- have "c \<noteq> cs" using assms
- using cs_waiting_def rest_nil waiting_eq wq_s_cs by auto
- from waiting_esI1[OF assms this]
- show ?thesis .
-qed
-
-lemma waiting_esE:
- assumes "waiting (e#s) t c"
- obtains "c \<noteq> cs" "waiting s t c"
-proof(cases "c = cs")
- case False
- hence "wq (e#s) c = wq s c" using is_v by auto
- with assms have "waiting s t c" using cs_waiting_def waiting_eq by auto
- from that(1)[OF False this] show ?thesis .
-next
- case True
- from no_waiting[OF assms[unfolded True]]
- show ?thesis by auto
-qed
-
-lemma holding_esE:
- assumes "holding (e#s) t c"
- obtains "c \<noteq> cs" "holding s t c"
-proof(cases "c = cs")
- case True
- from no_holding[OF assms[unfolded True]]
- show ?thesis by auto
-next
- case False
- hence "wq (e#s) c = wq s c" using is_v by auto
- from assms[unfolded s_holding_def, folded wq_def,
- unfolded this, unfolded wq_def, folded s_holding_def]
- have "holding s t c" .
- from that[OF False this] show ?thesis .
-qed
-
-end
-
-lemma rel_eqI:
- assumes "\<And> x y. (x,y) \<in> A \<Longrightarrow> (x,y) \<in> B"
- and "\<And> x y. (x,y) \<in> B \<Longrightarrow> (x, y) \<in> A"
- shows "A = B"
- using assms by auto
-
-lemma in_RAG_E:
- assumes "(n1, n2) \<in> RAG (s::state)"
- obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs"
- | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs"
- using assms[unfolded s_RAG_def, folded waiting_eq holding_eq]
- by auto
-
-context valid_trace_v
-begin
-
-lemma RAG_es:
- "RAG (e # s) =
- RAG s - {(Cs cs, Th th)} -
- {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
- {(Cs cs, Th th') |th'. next_th s th cs th'}" (is "?L = ?R")
-proof(rule rel_eqI)
- fix n1 n2
- assume "(n1, n2) \<in> ?L"
- thus "(n1, n2) \<in> ?R"
- proof(cases rule:in_RAG_E)
- case (waiting th' cs')
- show ?thesis
- proof(cases "rest = []")
- case False
- interpret h_n: valid_trace_v_n s e th cs
- by (unfold_locales, insert False, simp)
- from waiting(3)
- show ?thesis
- proof(cases rule:h_n.waiting_esE)
- case 1
- with waiting(1,2)
- show ?thesis
- by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
- fold waiting_eq, auto)
- next
- case 2
- with waiting(1,2)
- show ?thesis
- by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
- fold waiting_eq, auto)
- qed
- next
- case True
- interpret h_e: valid_trace_v_e s e th cs
- by (unfold_locales, insert True, simp)
- from waiting(3)
- show ?thesis
- proof(cases rule:h_e.waiting_esE)
- case 1
- with waiting(1,2)
- show ?thesis
- by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def,
- fold waiting_eq, auto)
- qed
- qed
- next
- case (holding th' cs')
- show ?thesis
- proof(cases "rest = []")
- case False
- interpret h_n: valid_trace_v_n s e th cs
- by (unfold_locales, insert False, simp)
- from holding(3)
- show ?thesis
- proof(cases rule:h_n.holding_esE)
- case 1
- with holding(1,2)
- show ?thesis
- by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
- fold waiting_eq, auto)
- next
- case 2
- with holding(1,2)
- show ?thesis
- by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
- fold holding_eq, auto)
- qed
- next
- case True
- interpret h_e: valid_trace_v_e s e th cs
- by (unfold_locales, insert True, simp)
- from holding(3)
- show ?thesis
- proof(cases rule:h_e.holding_esE)
- case 1
- with holding(1,2)
- show ?thesis
- by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def,
- fold holding_eq, auto)
- qed
- qed
- qed
-next
- fix n1 n2
- assume h: "(n1, n2) \<in> ?R"
- show "(n1, n2) \<in> ?L"
- proof(cases "rest = []")
- case False
- interpret h_n: valid_trace_v_n s e th cs
- by (unfold_locales, insert False, simp)
- from h[unfolded h_n.waiting_set_eq h_n.holding_set_eq]
- have "((n1, n2) \<in> RAG s \<and> (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th)
- \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)) \<or>
- (n2 = Th h_n.taker \<and> n1 = Cs cs)"
- by auto
- thus ?thesis
- proof
- assume "n2 = Th h_n.taker \<and> n1 = Cs cs"
- with h_n.holding_taker
- show ?thesis
- by (unfold s_RAG_def, fold holding_eq, auto)
- next
- assume h: "(n1, n2) \<in> RAG s \<and>
- (n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th) \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)"
- hence "(n1, n2) \<in> RAG s" by simp
- thus ?thesis
- proof(cases rule:in_RAG_E)
- case (waiting th' cs')
- from h and this(1,2)
- have "th' \<noteq> h_n.taker \<or> cs' \<noteq> cs" by auto
- hence "waiting (e#s) th' cs'"
- proof
- assume "cs' \<noteq> cs"
- from waiting_esI1[OF waiting(3) this]
- show ?thesis .
- next
- assume neq_th': "th' \<noteq> h_n.taker"
- show ?thesis
- proof(cases "cs' = cs")
- case False
- from waiting_esI1[OF waiting(3) this]
- show ?thesis .
- next
- case True
- from h_n.waiting_esI2[OF waiting(3)[unfolded True] neq_th', folded True]
- show ?thesis .
- qed
- qed
- thus ?thesis using waiting(1,2)
- by (unfold s_RAG_def, fold waiting_eq, auto)
- next
- case (holding th' cs')
- from h this(1,2)
- have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
- hence "holding (e#s) th' cs'"
- proof
- assume "cs' \<noteq> cs"
- from holding_esI2[OF this holding(3)]
- show ?thesis .
- next
- assume "th' \<noteq> th"
- from holding_esI1[OF holding(3) this]
- show ?thesis .
- qed
- thus ?thesis using holding(1,2)
- by (unfold s_RAG_def, fold holding_eq, auto)
- qed
- qed
- next
- case True
- interpret h_e: valid_trace_v_e s e th cs
- by (unfold_locales, insert True, simp)
- from h[unfolded h_e.waiting_set_eq h_e.holding_set_eq]
- have h_s: "(n1, n2) \<in> RAG s" "(n1, n2) \<noteq> (Cs cs, Th th)"
- by auto
- from h_s(1)
- show ?thesis
- proof(cases rule:in_RAG_E)
- case (waiting th' cs')
- from h_e.waiting_esI2[OF this(3)]
- show ?thesis using waiting(1,2)
- by (unfold s_RAG_def, fold waiting_eq, auto)
- next
- case (holding th' cs')
- with h_s(2)
- have "cs' \<noteq> cs \<or> th' \<noteq> th" by auto
- thus ?thesis
- proof
- assume neq_cs: "cs' \<noteq> cs"
- from holding_esI2[OF this holding(3)]
- show ?thesis using holding(1,2)
- by (unfold s_RAG_def, fold holding_eq, auto)
- next
- assume "th' \<noteq> th"
- from holding_esI1[OF holding(3) this]
- show ?thesis using holding(1,2)
- by (unfold s_RAG_def, fold holding_eq, auto)
- qed
- qed
- qed
-qed
-
-end
-
-lemma step_RAG_v:
-assumes vt:
- "vt (V th cs#s)"
-shows "
- RAG (V th cs # s) =
- RAG s - {(Cs cs, Th th)} -
- {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
- {(Cs cs, Th th') |th'. next_th s th cs th'}" (is "?L = ?R")
-proof -
- interpret vt_v: valid_trace_v s "V th cs"
- using assms step_back_vt by (unfold_locales, auto)
- show ?thesis using vt_v.RAG_es .
-qed
-
-lemma (in valid_trace_create)
- th_not_in_threads: "th \<notin> threads s"
-proof -
- from pip_e[unfolded is_create]
- show ?thesis by (cases, simp)
-qed
-
-lemma (in valid_trace_create)
- threads_es [simp]: "threads (e#s) = threads s \<union> {th}"
- by (unfold is_create, simp)
-
-lemma (in valid_trace_exit)
- threads_es [simp]: "threads (e#s) = threads s - {th}"
- by (unfold is_exit, simp)
-
-lemma (in valid_trace_p)
- threads_es [simp]: "threads (e#s) = threads s"
- by (unfold is_p, simp)
-
-lemma (in valid_trace_v)
- threads_es [simp]: "threads (e#s) = threads s"
- by (unfold is_v, simp)
-
-lemma (in valid_trace_v)
- th_not_in_rest[simp]: "th \<notin> set rest"
-proof
- assume otherwise: "th \<in> set rest"
- have "distinct (wq s cs)" by (simp add: wq_distinct)
- from this[unfolded wq_s_cs] and otherwise
- show False by auto
-qed
-
-lemma (in valid_trace_v)
- set_wq_es_cs [simp]: "set (wq (e#s) cs) = set (wq s cs) - {th}"
-proof(unfold wq_es_cs wq'_def, rule someI2)
- show "distinct rest \<and> set rest = set rest"
- by (simp add: distinct_rest)
-next
- fix x
- assume "distinct x \<and> set x = set rest"
- thus "set x = set (wq s cs) - {th}"
- by (unfold wq_s_cs, simp)
-qed
-
-lemma (in valid_trace_exit)
- th_not_in_wq: "th \<notin> set (wq s cs)"
-proof -
- from pip_e[unfolded is_exit]
- show ?thesis
- by (cases, unfold holdents_def s_holding_def, fold wq_def,
- auto elim!:runing_wqE)
-qed
-
-lemma (in valid_trace) wq_threads:
- assumes "th \<in> set (wq s cs)"
- shows "th \<in> threads s"
- using assms
-proof(induct rule:ind)
- case (Nil)
- thus ?case by (auto simp:wq_def)
-next
- case (Cons s e)
- interpret vt_e: valid_trace_e s e using Cons by simp
- show ?case
- proof(cases e)
- case (Create th' prio')
- interpret vt: valid_trace_create s e th' prio'
- using Create by (unfold_locales, simp)
- show ?thesis
- using Cons.hyps(2) Cons.prems by auto
- next
- case (Exit th')
- interpret vt: valid_trace_exit s e th'
- using Exit by (unfold_locales, simp)
- show ?thesis
- using Cons.hyps(2) Cons.prems vt.th_not_in_wq by auto
- next
- case (P th' cs')
- interpret vt: valid_trace_p s e th' cs'
- using P by (unfold_locales, simp)
- show ?thesis
- using Cons.hyps(2) Cons.prems readys_threads
- runing_ready vt.is_p vt.runing_th_s vt_e.wq_in_inv
- by fastforce
- next
- case (V th' cs')
- interpret vt: valid_trace_v s e th' cs'
- using V by (unfold_locales, simp)
- show ?thesis using Cons
- using vt.is_v vt.threads_es vt_e.wq_in_inv by blast
- next
- case (Set th' prio)
- interpret vt: valid_trace_set s e th' prio
- using Set by (unfold_locales, simp)
- show ?thesis using Cons.hyps(2) Cons.prems vt.is_set
- by (auto simp:wq_def Let_def)
- qed
-qed
-
-context valid_trace
-begin
-
-lemma dm_RAG_threads:
- assumes in_dom: "(Th th) \<in> Domain (RAG s)"
- shows "th \<in> threads s"
-proof -
- from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
- moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
- ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
- hence "th \<in> set (wq s cs)"
- by (unfold s_RAG_def, auto simp:cs_waiting_def)
- from wq_threads [OF this] show ?thesis .
-qed
-
-lemma cp_le:
- assumes th_in: "th \<in> threads s"
- shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
-proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def)
- show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}))
- \<le> Max ((\<lambda>th. preced th s) ` threads s)"
- (is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
- proof(rule Max_f_mono)
- show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}" by simp
- next
- from finite_threads
- show "finite (threads s)" .
- next
- from th_in
- show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> threads s"
- apply (auto simp:Domain_def)
- apply (rule_tac dm_RAG_threads)
- apply (unfold trancl_domain [of "RAG s", symmetric])
- by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def)
- qed
-qed
-
-lemma max_cp_eq:
- shows "Max ((cp s) ` threads s) = Max ((\<lambda> th. (preced th s)) ` threads s)"
- (is "?l = ?r")
-proof(cases "threads s = {}")
- case True
- thus ?thesis by auto
-next
- case False
- have "?l \<in> ((cp s) ` threads s)"
- proof(rule Max_in)
- from finite_threads
- show "finite (cp s ` threads s)" by auto
- next
- from False show "cp s ` threads s \<noteq> {}" by auto
- qed
- then obtain th
- where th_in: "th \<in> threads s" and eq_l: "?l = cp s th" by auto
- have "\<dots> \<le> ?r" by (rule cp_le[OF th_in])
- moreover have "?r \<le> cp s th" (is "Max (?f ` ?A) \<le> cp s th")
- proof -
- have "?r \<in> (?f ` ?A)"
- proof(rule Max_in)
- from finite_threads
- show " finite ((\<lambda>th. preced th s) ` threads s)" by auto
- next
- from False show " (\<lambda>th. preced th s) ` threads s \<noteq> {}" by auto
- qed
- then obtain th' where
- th_in': "th' \<in> ?A " and eq_r: "?r = ?f th'" by auto
- from le_cp [of th'] eq_r
- have "?r \<le> cp s th'"
- moreover have "\<dots> \<le> cp s th"
- proof(fold eq_l)
- show " cp s th' \<le> Max (cp s ` threads s)"
- proof(rule Max_ge)
- from th_in' show "cp s th' \<in> cp s ` threads s"
- by auto
- next
- from finite_threads
- show "finite (cp s ` threads s)" by auto
- qed
- qed
- ultimately show ?thesis by auto
- qed
- ultimately show ?thesis using eq_l by auto
-qed
-
-lemma max_cp_eq_the_preced:
- shows "Max ((cp s) ` threads s) = Max (the_preced s ` threads s)"
- using max_cp_eq using the_preced_def by presburger
-
-end
-
-lemma preced_v [simp]: "preced th' (V th cs#s) = preced th' s"
- by (unfold preced_def, simp)
-
-lemma (in valid_trace_v)
- preced_es: "preced th (e#s) = preced th s"
- by (unfold is_v preced_def, simp)
-
-lemma the_preced_v[simp]: "the_preced (V th cs#s) = the_preced s"
-proof
- fix th'
- show "the_preced (V th cs # s) th' = the_preced s th'"
- by (unfold the_preced_def preced_def, simp)
-qed
-
-lemma (in valid_trace_v)
- the_preced_es: "the_preced (e#s) = the_preced s"
- by (unfold is_v preced_def, simp)
-
-context valid_trace_p
-begin
-
-lemma not_holding_es_th_cs: "\<not> holding s th cs"
-proof
- assume otherwise: "holding s th cs"
- from pip_e[unfolded is_p]
- show False
- proof(cases)
- case (thread_P)
- moreover have "(Cs cs, Th th) \<in> RAG s"
- using otherwise cs_holding_def
- holding_eq th_not_in_wq by auto
- ultimately show ?thesis by auto
- qed
-qed
-
-lemma waiting_kept:
- assumes "waiting s th' cs'"
- shows "waiting (e#s) th' cs'"
- using assms
- by (metis cs_waiting_def hd_append2 list.sel(1) list.set_intros(2)
- rotate1.simps(2) self_append_conv2 set_rotate1
- th_not_in_wq waiting_eq wq_es_cs wq_neq_simp)
-
-lemma holding_kept:
- assumes "holding s th' cs'"
- shows "holding (e#s) th' cs'"
-proof(cases "cs' = cs")
- case False
- hence "wq (e#s) cs' = wq s cs'" by simp
- with assms show ?thesis using cs_holding_def holding_eq by auto
-next
- case True
- from assms[unfolded s_holding_def, folded wq_def]
- obtain rest where eq_wq: "wq s cs' = th'#rest"
- by (metis empty_iff list.collapse list.set(1))
- hence "wq (e#s) cs' = th'#(rest@[th])"
- by (simp add: True wq_es_cs)
- thus ?thesis
- by (simp add: cs_holding_def holding_eq)
-qed
-
-end
-
-locale valid_trace_p_h = valid_trace_p +
- assumes we: "wq s cs = []"
-
-locale valid_trace_p_w = valid_trace_p +
- assumes wne: "wq s cs \<noteq> []"
-begin
-
-definition "holder = hd (wq s cs)"
-definition "waiters = tl (wq s cs)"
-definition "waiters' = waiters @ [th]"
-
-lemma wq_s_cs: "wq s cs = holder#waiters"
- by (simp add: holder_def waiters_def wne)
-
-lemma wq_es_cs': "wq (e#s) cs = holder#waiters@[th]"
- by (simp add: wq_es_cs wq_s_cs)
-
-lemma waiting_es_th_cs: "waiting (e#s) th cs"
- using cs_waiting_def th_not_in_wq waiting_eq wq_es_cs' wq_s_cs by auto
-
-lemma RAG_edge: "(Th th, Cs cs) \<in> RAG (e#s)"
- by (unfold s_RAG_def, fold waiting_eq, insert waiting_es_th_cs, auto)
-
-lemma holding_esE:
- assumes "holding (e#s) th' cs'"
- obtains "holding s th' cs'"
- using assms
-proof(cases "cs' = cs")
- case False
- hence "wq (e#s) cs' = wq s cs'" by simp
- with assms show ?thesis
- using cs_holding_def holding_eq that by auto
-next
- case True
- with assms show ?thesis
- by (metis cs_holding_def holding_eq list.sel(1) list.set_intros(1) that
- wq_es_cs' wq_s_cs)
-qed
-
-lemma waiting_esE:
- assumes "waiting (e#s) th' cs'"
- obtains "th' \<noteq> th" "waiting s th' cs'"
- | "th' = th" "cs' = cs"
-proof(cases "waiting s th' cs'")
- case True
- have "th' \<noteq> th"
- proof
- assume otherwise: "th' = th"
- from True[unfolded this]
- show False by (simp add: th_not_waiting)
- qed
- from that(1)[OF this True] show ?thesis .
-next
- case False
- hence "th' = th \<and> cs' = cs"
- by (metis assms cs_waiting_def holder_def list.sel(1) rotate1.simps(2)
- set_ConsD set_rotate1 waiting_eq wq_es_cs wq_es_cs' wq_neq_simp)
- with that(2) show ?thesis by metis
-qed
-
-lemma RAG_es: "RAG (e # s) = RAG s \<union> {(Th th, Cs cs)}" (is "?L = ?R")
-proof(rule rel_eqI)
- fix n1 n2
- assume "(n1, n2) \<in> ?L"
- thus "(n1, n2) \<in> ?R"
- proof(cases rule:in_RAG_E)
- case (waiting th' cs')
- from this(3)
- show ?thesis
- proof(cases rule:waiting_esE)
- case 1
- thus ?thesis using waiting(1,2)
- by (unfold s_RAG_def, fold waiting_eq, auto)
- next
- case 2
- thus ?thesis using waiting(1,2) by auto
- qed
- next
- case (holding th' cs')
- from this(3)
- show ?thesis
- proof(cases rule:holding_esE)
- case 1
- with holding(1,2)
- show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
- qed
- qed
-next
- fix n1 n2
- assume "(n1, n2) \<in> ?R"
- hence "(n1, n2) \<in> RAG s \<or> (n1 = Th th \<and> n2 = Cs cs)" by auto
- thus "(n1, n2) \<in> ?L"
- proof
- assume "(n1, n2) \<in> RAG s"
- thus ?thesis
- proof(cases rule:in_RAG_E)
- case (waiting th' cs')
- from waiting_kept[OF this(3)]
- show ?thesis using waiting(1,2)
- by (unfold s_RAG_def, fold waiting_eq, auto)
- next
- case (holding th' cs')
- from holding_kept[OF this(3)]
- show ?thesis using holding(1,2)
- by (unfold s_RAG_def, fold holding_eq, auto)
- qed
- next
- assume "n1 = Th th \<and> n2 = Cs cs"
- thus ?thesis using RAG_edge by auto
- qed
-qed
-
-end
-
-context valid_trace_p_h
-begin
-
-lemma wq_es_cs': "wq (e#s) cs = [th]"
- using wq_es_cs[unfolded we] by simp
-
-lemma holding_es_th_cs:
- shows "holding (e#s) th cs"
-proof -
- from wq_es_cs'
- have "th \<in> set (wq (e#s) cs)" "th = hd (wq (e#s) cs)" by auto
- thus ?thesis using cs_holding_def holding_eq by blast
-qed
-
-lemma RAG_edge: "(Cs cs, Th th) \<in> RAG (e#s)"
- by (unfold s_RAG_def, fold holding_eq, insert holding_es_th_cs, auto)
-
-lemma waiting_esE:
- assumes "waiting (e#s) th' cs'"
- obtains "waiting s th' cs'"
- using assms
- by (metis cs_waiting_def event.distinct(15) is_p list.sel(1)
- set_ConsD waiting_eq we wq_es_cs' wq_neq_simp wq_out_inv)
-
-lemma holding_esE:
- assumes "holding (e#s) th' cs'"
- obtains "cs' \<noteq> cs" "holding s th' cs'"
- | "cs' = cs" "th' = th"
-proof(cases "cs' = cs")
- case True
- from held_unique[OF holding_es_th_cs assms[unfolded True]]
- have "th' = th" by simp
- from that(2)[OF True this] show ?thesis .
-next
- case False
- have "holding s th' cs'" using assms
- using False cs_holding_def holding_eq by auto
- from that(1)[OF False this] show ?thesis .
-qed
-
-lemma RAG_es: "RAG (e # s) = RAG s \<union> {(Cs cs, Th th)}" (is "?L = ?R")
-proof(rule rel_eqI)
- fix n1 n2
- assume "(n1, n2) \<in> ?L"
- thus "(n1, n2) \<in> ?R"
- proof(cases rule:in_RAG_E)
- case (waiting th' cs')
- from this(3)
- show ?thesis
- proof(cases rule:waiting_esE)
- case 1
- thus ?thesis using waiting(1,2)
- by (unfold s_RAG_def, fold waiting_eq, auto)
- qed
- next
- case (holding th' cs')
- from this(3)
- show ?thesis
- proof(cases rule:holding_esE)
- case 1
- with holding(1,2)
- show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
- next
- case 2
- with holding(1,2) show ?thesis by auto
- qed
- qed
-next
- fix n1 n2
- assume "(n1, n2) \<in> ?R"
- hence "(n1, n2) \<in> RAG s \<or> (n1 = Cs cs \<and> n2 = Th th)" by auto
- thus "(n1, n2) \<in> ?L"
- proof
- assume "(n1, n2) \<in> RAG s"
- thus ?thesis
- proof(cases rule:in_RAG_E)
- case (waiting th' cs')
- from waiting_kept[OF this(3)]
- show ?thesis using waiting(1,2)
- by (unfold s_RAG_def, fold waiting_eq, auto)
- next
- case (holding th' cs')
- from holding_kept[OF this(3)]
- show ?thesis using holding(1,2)
- by (unfold s_RAG_def, fold holding_eq, auto)
- qed
- next
- assume "n1 = Cs cs \<and> n2 = Th th"
- with holding_es_th_cs
- show ?thesis
- by (unfold s_RAG_def, fold holding_eq, auto)
- qed
-qed
-
-end
-
-context valid_trace_p
-begin
-
-lemma RAG_es': "RAG (e # s) = (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
- else RAG s \<union> {(Th th, Cs cs)})"
-proof(cases "wq s cs = []")
- case True
- interpret vt_p: valid_trace_p_h using True
- by (unfold_locales, simp)
- show ?thesis by (simp add: vt_p.RAG_es vt_p.we)
-next
- case False
- interpret vt_p: valid_trace_p_w using False
- by (unfold_locales, simp)
- show ?thesis by (simp add: vt_p.RAG_es vt_p.wne)
-qed
-
-end
-
-
-end